Calculus .1 Limit of a Function

4. Iterative Method

1.4 Calculus .1 Limit of a Function

Let y = f(x)

Then limf(x)= i.e, “ f(x) as x a” implies for any (>0), (>0) such that whenever 0< |x a|< , |f(x) |<

Some Standard Expansions

(1 x) = 1 nx ^{( )} n(n 1)(n )

x . . . x x a

x a = x xa xa . . . a e = 1 + x +

. . . log(1 x) = x + . . . log(1 x) = x . . . Sin x = x

. . . Cos x = 1

. . . Sinh x = x

. . . Cosh x = 1 + . . . Some Important Limits

lim sinx

x =

lim(1 1

x ) =

lim(1 x) =

lim a 1

x = log a lime 1

x = 1

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lim

log(1 x)

x = 1

lim x a

x a = a

limlog|x| = L – Hospital’s ule

 When function is of or form, differentiate numerator & denominator and then apply limit.

Existence of Limits and Continuity:

1. f(x) is defined at a, i.e, f(a) exists.

2. If lim

f(x) = lim

f(x) = L ,then the lim

f(x) exists and equal to L.

3. If limf(x) = limf(x)= f(a) then the function f(x) is said to be continuous.

Properties of Continuity

If f and g are two continuous functions at a; then a. (f+g), (f.g), (f-g) are continuous at a b. is continuous at a, provided g(a) 0 c. |f| or |g| is continuous at a

olle’s theorem

If (i) f(x) is continuous in closed interval [a,b]

(ii) f’(x) exists for every value of x in open interval (a,b) (iii) f(a) = f(b)

Then there exists at least one point c between (a, b) such that ( ) = 0

Geometrically: There exists at least one point c between (a, b) such that tangent at c is parallel to x axis

Lagrange’s Mean Value Theorem

If (i) f(x) is continuous in the closed interval [a,b] and

(ii) f’(x) exists in the open interval (a,b), then atleast one value c of x exist in (a,b) such that ^{( ) ( )}

= f (c).

Geometrically, it means that at point c, tangent is parallel to the chord line.

Cauchy’s Mean Value Theorem

If (i) f(x) is continuous in the closed interval [a,a+h] and

(ii) f (x) exists in the open interval (a,a+h), then there is at least one number (0< <1) such that

f(a+h) = f(a) + h f(a+ h) Let f1 and f2 be two functions:

i) f1,f2 both are continuous in [a,b]

ii) f1, f2 both are differentiable in (a,b) iii) f2’ 0 in (a,b)

then, for a b

( ) ( ) ( ) ( ) = ^{( )}

( )

1.4.2 Derivative:

’( ) = lim ( ) ( )

Provided the limit exists ’( ) is called the rate of change of f at x.

Algebra of derivative:-

i. (f g) = f g ii. (f g) = f – g iii. (f. g) = f . g f . g iv. (f/g) = ^.^.

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Monotonicity of a Function f(x)

1. f(x) is increasing function if for , f( ) f( ) Necessary and sufficient condition, f’ (x) 2. f(x) is decreasing function if for , , f( ) f( )

Necessary and sufficient condition, f (x)

Note: If f is a monotonic function on a domain ‘D’ then f is one-one on D.

Maxima-Minima a) Global b) Local

Rule for finding maxima & minima:

 If maximum or minimum value of f(x) is to be found, let y = f(x)

If = , proceed further and find at x = .

If , y has neither maximum nor minimum value at x = But If

= , proceed further and find

at x = . If , y has minimum value

If , y has maximum value If = , proceed further

Note: Greatest / least value exists either at critical point or at the end point of interval.

Point of Inflexion

If at a point, the following conditions are met, then such point is called point of inflexion

= ,

ii) = 0 ,

iii)

 Neither minima nor maxima exists Taylor Series:

f(a h)= f(a) h f’(a)

f”(a) . . . Maclaurian Series:

f(x) = f( ) x f’( ) f ( ) ^hf ( ) Maxima & Minima (Two variables)

r =

, s =

, t =

= 0,

= solve these equations. Let the solution be (a, b), (c, d)…

2. (i) if rt s and r maximum at (a, b) (ii) if rt s and r minimum at (a, b)

(iii) if rt s < 0 at (a, b), f(a,b) is not an extreme value i.e, f(a, b) is saddle point.

(iv) if rt s > 0 at (a, b), It is doubtful, need further investigation.

Point of inflexion

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39. ∫ e ,f(x) f (x)-dx = e f(x)

Integration by parts: ∫ u v dx = u. ∫ v dx ∫( ∫ v dx)dx

1.4.5 Rules for Definite Integral

1. ∫ f(x)dx =∫ f(x)dx+∫ f(x)dx a<c<b

2. ∫ f(x)dx =∫ f(a b x)dx ∫ f(x)dx =∫ f(a x)dx

3. ∫ f(x)dx =∫^/f(x)dx+∫^/f(a x)dx ∫ f(x)dx = ∫^/f(x)dx if f(a-x)=f(x)

= 0 if f(a-x)=-f(x)

4. ∫ f(x)dx =2 ∫ f(x)dx if f(-x) = f(x), even function = 0 if f(x) = -f(x), odd function Improper Integral

Those integrals for which limit is infinite or integrand is infinite in a x b in case of ∫ f(x)dx, then it is called as improper integral.

1.4.6 Convergence:

 ∫ f(x)dx is said to be convergent if the value of the integral is finite.

 If (i) f(x) g(x) for all x and (ii) ∫ g(x)dx converges , then ∫ f(x)dx also converges

 If (i) f(x) g(x) for all x and (ii) ∫ g(x)dx diverges, then ∫ f(x)dx also diverges

 If lim^{( )}

( ) = c where c 0, then both integrals ∫ f(x)dx and ∫ g(x)dx converge or both diverge.

 ∫ is converges when p 1 and diverges when p 1

 ∫ edx and ∫edx is converges for any constant p and diverges for p

 The integral ∫

( ) is convergent if and only if p 1

 The integral ∫

( ) is convergent if and only if p 1 Selection of U & V I L A T E

Logarithmic Inverse circular (e.g. tan¹x)

Algebraic Trigonometric

Exponential

Note: Take that function as “u” which comes first in “ILATE”

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1.4.7 Vector Calculus:

Scalar Point Function:

If corresponding to each point P of region R there is a corresponding scalar then (P) is said to be a scalar point function for the region R.

(P)= (x,y,z)

Vector Point Function:

If corresponding to each point P of region R, there corresponds a vector defined by F(P) then F is called a vector point function for region R.

F(P) = F(x,y,z) = f1(x,y,z) ̂ +f2(x,y,z)ĵ f3(x,y,z) ̂ Vector Differential Operator or Del Operator: = .

/ Directional Derivative:

The directional derivative of f in a direction N⃗⃗ is the resolved part of f in direction N⃗⃗ . f. N⃗⃗ = | f|cos

Where N⃗⃗ is a unit vector in a particular direction.

Direction cosine: l m n = 1 Where, l =cos , m=cos , n=cos , 1.4.8 Gradient:

The vector function f is defined as the gradient of the scalar point function f(x,y,z) and written as grad f.

grad f = f = î

+ ̂

 f is vector function

 If f(x,y,z) = 0 is any surface, then f is a vector normal to the surface f and has a magnitude equal to rate of change of f along this normal.

 Directional derivative of f(x,y,z) is maximum along f and magnitude of this maximum is | f|.

1.4.9 Divergence:

The divergence of a continuously differentiable vector point function F is denoted by div. F and is defined by the equation.

div. F = . F

F = f + ĵ Ψ ̂ 1.4.11 Solenoidal Vector Function

If .A = 0 , then A is called as solenoidal vector function.

1.4.12 Irrotational Vector Function

If A =0, then A is said to be irrotational otherwise rotational.

1.4.13 DEL Applied Twice to Point Functions:

1. div grad f = f=

f, g are scalar functions & F, G are vector functions 1. (f g) = f + g

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9. (F G) = F( G) G( F) Also note:

1. (f/g)= (g f – f g)/g 2. (F.G)’ = F’.G F . G’

3. (F G)’ = F’ G + F G’

4. (fg) = g f + 2 f. g + f g

1.4.15 Vector product

1. Dot product of A B with C is called scalar triplet product and denoted as [ABC]

Rule: For evaluating the scalar triplet product (i) Independent of position of dot and cross (ii) Dependent on the cyclic order of the vector [ABC] = A B. C = A. B C

= B C. A= B.C A = C A. B = C.A B A B. C = -(B A. C)

2. (A⃗⃗ B⃗⃗ ) C⃗ = (extreme adjacent) Outer

= (Outer. extreme) adjacent (Outer. adjacent) extreme

 (A ⃗⃗⃗ B)⃗⃗⃗⃗ C⃗ = (C⃗ . A⃗⃗ ) B⃗⃗ - (C⃗ . B⃗⃗ ) A⃗⃗

 A⃗⃗ (B⃗⃗ C⃗ ) = (A⃗⃗ . C⃗ ) B⃗⃗ - (A⃗⃗ . B⃗⃗ ) C⃗

 (A⃗⃗ B⃗⃗ ) C⃗ A⃗⃗ (B⃗⃗ C⃗ )

1.4.16 Line Integral, Surface Integral & Volume Integral

 Line integral = ∫ F( )d

If F( )= f(x,y,z) ĵ (x,y,z) + ̂ Ψ(x,y,z) d = dx ĵ dy ̂ dz

∫ F( )d = ∫ ( f dx dy Ψ dz )

 Surface integral: ∫ F⃗ . ds⃗⃗⃗⃗ or ∫ F⃗ . N⃗⃗ ds, Where N is unit outward normal to Surface.

 Volume integral : ∫ Fdv

If F(R ) = f(x,y,z)î + (x,y,z)ĵ Ψ (x,y,z) ̂ and v = x y z , then

∫ F dv = î∫ ∫ ∫ fdxdydz ĵ ∫ ∫ ∫ dxdydz + ̂ ∫ ∫ ∫ Ψdxdydz 1.4.17 Green’s Theorem

If R be a closed region in the xy plane bounded by a simple closed curve c and if P and Q are continuous functions of x and y having continuous derivative in , then according to Green’s theorem.

∮ (P dx dy) = ∫ ∫ .

/ dxdy

1.4.18 Sto e’s theorem

If F be continuously differentiable vector function in R, then ∮ F. dr = ∫ F .N ds 1.4.19 Gauss divergence theorem

The normal surface integral of a vector point function F which is continuously differentiable over the boundary of a closed region is equal to the

∫ F .N.ds =∫ div F dv

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1.5: Differential Equations

1.5.1 Order of Differential Equation: It is the order of the highest derivative appearing in it.

1.5.2 Degree of Differential Equation: It is the degree of the highest derivative occurring in it, after expressing the equation free from radicals and fractions as far as derivatives are concerned.

1.5.3 Differential Equations of First Order First Degree:

Equations of first order and first degree can be expressed in the form f (x, y, y ) = or y = f(x, y). Following are the different ways of solving equations of first order and first degree:

1. Variable separable : f(x)dx + g(y)dy = 0

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